Istituto di Biofisica     
Bologna M., Tsallis C., Grigolini P. Anomalous diffusion associated with nonlinear fractional derivative Fokker-Planck-like equation: Exact time-dependent solutions. In: Physical Review E, vol. 62 (2) pp. 2213 - 2218. American Physical Society, 2000.
We consider the d=1 nonlinear Fokker-Planck-like equation with fractional derivatives (∂/∂t)P(x,t)=D(∂γ/∂xγ)[P(x,t)]ν. Exact time-dependent solutions are found for ν=(2-γ)/(1+γ)(-∞<γ<~2). By considering the long-distance asymptotic behavior of these solutions, a connection is established, namely, q=(γ+3)/(γ+1)(0<γ<~2), with the solutions optimizing the nonextensive entropy characterized by index q. Interestingly enough, this relation coincides with the one already known for LÚvy-like superdiffusion (i.e., ν=1 and 0<γ<~2). Finally, for (γ,ν)=(2,0) we obtain q=5/3, which differs from the value q=2 corresponding to the γ=2 solutions available in the literature (ν<1 porous medium equation), thus exhibiting nonuniform convergence.
DOI: 10.1103/PhysRevE.62.2213
Subject Nonlinear fractional derivativ

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